On divisor sums due to Erdős and Ramanujan

Abstract: Let $d(n)$ denote the number of divisors of a positive integer $n$. A classical problem in analytic number theory is given by the asymptotic behavior of the divisor sum $\sum_{n \leq x} \frac{1}{d(n)}$, with Ramanujan having introduced an asymptotic formula for this sum with an explicit evaluation for the constant $A_1$ for the leading term $A_1 \frac{x}{\sqrt{\log x}}$. Gabdullin et al. recently considered a hybrid of this problem and the Titchmarsh divisor problem concerning $\sum_{p\leq x} d(p-1)$, proving that $$\sum_{p\leq x} \frac{1}{d(p-1)} \asymp \frac{x}{(\log x)^{3/2}}.$$ This result, together with Erdős's asymptotic formula $\sum_{n \leq x} d(d(n)) \sim c \, x \log \log x $ for a constant $c \in (0, \infty)$, lead us to consider the hybrid $\sum_{n \leq x} \frac{1}{d(d(n))}$ of the Erdős and Ramanujan divisor sums. The presence of the reciprocal significantly complicates the analysis, as it amplifies the contribution of integers for which $d(d(n))$ is exceptionally small. In this paper, we prove that $$\sum_{n \leq x} \frac{1}{d(d(n))} \asymp \frac{x}{ \log \log x}, $$ through a combined application of Golomb's estimate for powerful numbers and Turán's quantitative form of the Hardy-Ramanujan theorem.

• The paper establishes that the hybrid divisor sum 1/d(d(n)) grows in order of x/log log x, providing matching upper and lower bounds.
• It employs advanced techniques such as decomposing numbers into squarefree and powerful parts and leveraging the Hardy–Ramanujan theorem.
• The work synthesizes classical divisor sum problems, extending Ramanujan's estimates and Erdős's iterated function analysis into a new hybrid framework.

Asymptotics of Hybrid Divisor Sums: Analysis of $\sum_{n \leq x} \frac{1}{d(d(n))}$

Introduction and Context

The paper "On divisor sums due to Erdős and Ramanujan" (2605.00695) focuses on a precise asymptotic order for a hybrid divisor sum of the form $\sum_{n \leq x} \frac{1}{d(d(n))}$, where $d(n)$ is the standard divisor function and $d(d(n))$ its second iterate. This sum acts as an overview of two classical lines of inquiry: divisor sum asymptotics originated by Ramanujan and the behavior of iterated divisor functions analyzed by Erdős. The main result provides a matching upper and lower bound, establishing that the sum is of order $x / \log \log x$.

The motivation lies in several landmark divisor sum problems:

• Ramanujan's estimate for $\sum_{n \leq x} 1/d(n)$, proved to be $x / (\log x)^{1/2}$ up to an explicit constant,
• Erdős's result that $\sum_{n \leq x} d(d(n)) \sim c_1 x \log\log x$ for some $c_1>0$,
• Recent work on sum-analogues involving prime arguments and hybrid forms, such as Gabdullin et al. for $\sum_{p \leq x} 1/d(p-1)$.

The investigation explores how the reciprocal operation in divisor sums, when composed with an additional layer of iterated divisors, creates an arithmetic function with significant irregularity and magnitude reduction compared to the non-reciprocal, non-iterated cases.

Main Results and Techniques

The principal theorem established is:

$\sum_{n \leq x} \frac{1}{d(d(n))}$0

This result is structurally parallel to the aforementioned Dirichlet, Ramanujan, and Erdős divisor sum results but occupies a distinct analytic setting due to the presence of both function iteration and reciprocal, which amplify the contribution of exceptional (atypical) values of $\sum_{n \leq x} \frac{1}{d(d(n))}$1.

Key elements of the proof include:

• Decomposition by integer structure: The analysis employs a factoring of each $\sum_{n \leq x} \frac{1}{d(d(n))}$2 as $\sum_{n \leq x} \frac{1}{d(d(n))}$3 where $\sum_{n \leq x} \frac{1}{d(d(n))}$4 is the squarefree part and $\sum_{n \leq x} \frac{1}{d(d(n))}$5 is the powerful part. This enables the use of precise counting and structural results on both squarefree and powerful numbers.
• Golomb’s bound governs the count of powerful numbers, yielding optimal estimates for sets where the powerful part becomes large relative to $\sum_{n \leq x} \frac{1}{d(d(n))}$6.
• Application of the Hardy–Ramanujan theorem (and Turán's variant): The normal order of $\sum_{n \leq x} \frac{1}{d(d(n))}$7 allows the isolation of an exceptional set where $\sum_{n \leq x} \frac{1}{d(d(n))}$8 can be particularly small, showing this set is negligible in total contribution to the sum.
• Lower bound via squarefree numbers: A matching lower bound is constructed by restricting to squarefree $\sum_{n \leq x} \frac{1}{d(d(n))}$9, for which the computation of $d(n)$0 simplifies and can be tightly controlled.

Upper Bound Structure: The set $d(n)$1 is partitioned into three disjoint classes based on the size of the powerful component $d(n)$2 and the count of distinct prime divisors $d(n)$3:

• Those with large $d(n)$4,
• Those with abnormal $d(n)$5,
• The generic case.

Each is shown to yield at most $d(n)$6 to the sum.

Lower Bound Construction: Focus on the set of squarefree $d(n)$7 with $d(n)$8 restricted to a typical range supplies an $d(n)$9 lower bound, exploiting the fact that for such $d(d(n))$0, $d(d(n))$1 simplifies to $d(d(n))$2, controllable in size.

Comparison with Prior Work and Notable Claims

This result is the natural hybrid of earlier work, synthesizing the complexity of the iterated divisor function with the dampening effect of the reciprocal.

• Order of magnitude: The $d(d(n))$3 result is in sharp contrast with the $d(d(n))$4 behavior of Erdős's direct sum and the $d(d(n))$5 of the Ramanujan sum.
• Parallel with recent prime-based results: The result matches, in structural form, the recent evaluation of $d(d(n))$6 by Gabdullin et al., both reflecting the irregularity introduced by composition and reciprocation in divisor sums.

Conjecture: In analogy to both Erdős and Gabdullin–Konyagin–Iudelevich, it is conjectured that there is in fact a true asymptotic,

$d(d(n))$7

for some explicit $d(d(n))$8. The paper regards sharp determination of $d(d(n))$9 as a hard problem, given the irregular and non-multiplicative nature of $x / \log \log x$0 and the accentuation of rare values by reciprocation.

Implications and Theoretical Significance

The main theorem illustrates the intricate behavior encountered in composed and inverted arithmetic functions, extending classical results into a regime where standard analytic number theory techniques require significant adaptation—primarily through the management of exceptional sets (through Turán’s refinement of the Hardy-Ramanujan theorem) and the use of precise counting techniques for powerful numbers.

The result further highlights the informational content lost when taking function iterates and reciprocals: the combined effect is to flatten the sum's growth rate substantially. It also underlines the challenge in extending divisor sum asymptotic evaluations—where explicit constants and finer secondary terms become substantially more elusive as one layers additional algebraic structure.

From a theoretical standpoint, this work places new hybrid sums within the established taxonomy of divisor summation problems and provides templates for analogous investigations, both in strengthening lower-order terms and in shifting perspective to sums over more restricted sets (e.g., over primes or integers with specific multiplicative constraints).

Practical impact is largely indirect, but any subsequent improvements in upper or lower bounds or error terms for such sums will feed into the precision of sieve-theoretic estimates, probabilistic models for integer factorization, and—ultimately—into computational algorithms relying on fast estimation of averages of arithmetic functions.

Conclusion

The paper rigorously establishes the asymptotic order $x / \log \log x$1, extending central themes in analytic number theory regarding divisor sums to new, structurally complex hybrids. The analytical approach is notable both for its elegant management of rare events and for synthesizing ideas from decomposition theory, probabilistic number theory, and advanced divisor analysis. The conjectured true asymptotic with a constant $x / \log \log x$2 remains an open, challenging direction. The techniques and insights here will inform both theoretical understanding and methodological development for related problems in divisor function iterates and their average behaviors.